One source of compact quantum groups is provided by ordinary compact Lie groups deformed in some sense ([1]}, [2]} and references therein), but more exotic examples can be obtained as quantum automorphism groups of various structures, such as, say, bilinear forms [3]}, finite-dimensional \(C^*\) -algebras endowed with a trace [4]}, [5]}, finite graphs [6]}, finite metric spaces [7]}, etc. In the context of simplicity, it turns out [8]} that deforming simple compact Lie groups produces, as expected, simple compact quantum groups. On the other hand, it is shown in the same paper (\(§\) 4) that quantum automorphism groups of traced finite-dimensional \(C^*\) -algebras are simple, while quantum automorphism groups of non-degenerate bilinear forms have a single non-trivial normal compact quantum subgroup of order 2.
